# Penelope Gehring

### Doktorandin

Kontakt
Raum:
2.09.3.15
Telefon:
+49 331 977-1847

#### Research interest

• Boundary value problems for Dirac operators on Lorentzian manifolds with timelike boundary
• Asymptotically hyperbolic Riemannian manifolds in context of General Relativity

#### Funding

PhD student of the IMPRS for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory

#### Publications

2022 | The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions | Christian Bär and Penelope GehringLink zum Preprint

### The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions

#### Autoren: Christian Bär and Penelope Gehring (2022)

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well under\-stood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.

2021 | Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass | Armando J. Cabrera Pacheco, Carla Cederbaum, Penelope Gehring, Alejandro Peñuela DiazLink zum Preprint

### Constructing electrically charged Riemannian manifolds with minimal boundary, prescribed asymptotics, and controlled mass

#### Autoren: Armando J. Cabrera Pacheco, Carla Cederbaum, Penelope Gehring, Alejandro Peñuela Diaz (2021)

In 2015, Mantoulidis and Schoen constructed 3-dimensional asymptotically Euclidean manifolds with non-negative scalar curvature whose ADM mass can be made arbitrarily close to the optimal value of the Riemannian Penrose Inequality, while the intrinsic geometry of the outermost minimal surface can be "far away" from being round. The resulting manifolds, called extensions, are geometrically not "close" to a spatial Schwarzschild manifold. This suggests instability of the Riemannian Penrose Inequality. Their construction was later adapted to $$n+1$$ dimensions by Cabrera Pacheco and Miao, suggesting instability of the higher dimensional Riemannian Penrose Inequality. In recent papers by Alaee, Cabrera Pacheco, and Cederbaum and by Cabrera Pacheco, Cederbaum, and McCormick, a similar construction was performed for asymptotically Euclidean, electrically charged Riemannian manifolds and for asymptotically hyperbolic Riemannian manifolds, respectively, obtaining 3-dimensional extensions that suggest instability of the Riemannian Penrose Inequality with electric charge and of the conjectured asymptotically hyperbolic Riemannian Penrose Inequality in 3 dimensions. This paper combines and generalizes all the aforementioned results by constructing suitable asymptotically hyperbolic or asymptotically Euclidean extensions with electric charge in $$n+1$$ dimensions for $$n \geq 2$$.
Besides suggesting instability of a naturally conjecturally generalized Riemannian Penrose Inequality, the constructed extensions give insights into an ad hoc generalized notion of Bartnik mass, similar to the Bartnik mass estimate for minimal surfaces proven by Mantoulidis and Schoen via their extensions, and unifying the Bartnik mass estimates in the various scenarios mentioned above.

2019 | The Isoperimetric Inequality: Proofs by Convex and Differential Geometry | Penelope GehringZeitschrift: Rose Hulman Undergraduate Mathematics JournalBand: 20, no. 2Link zur Publikation

### The Isoperimetric Inequality: Proofs by Convex and Differential Geometry

#### Autoren: Penelope Gehring (2019)

The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of elementary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn--Minkowski-Inequality. Using this inequality, the Isoperimetric Inquality in n dimensions will be shown.

Zeitschrift:
Rose Hulman Undergraduate Mathematics Journal
Band:
20, no. 2